Abstract: We establish the result of the title.
In combinatorial terms this has the implication that for sufficiently small
eps > 0, for all n, any marking of an eps fraction of the vertices of the
n-dimensional hypercube necessarily leaves a vertex x such that marked vertices
are a minority of every sphere centered at x.

Comments:

17 pages. v2. This version matches the published version and fixes typos, simplifies some proofs and makes other minor changes. The main results are unchanged from v1